### A note on axiomatisations of two-dimensional modal logics.A note on axiomatisations of two-dimensional modal logics.

Access Restriction
Open

 Author Kurucz, Agi Source CiteSeerX Content type Text File Format PDF
 Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science Subject Keyword Two-dimensional Modal Logic ♦ Finite Variable Fragment ♦ Modal Axiom ♦ Finite Dimensional Cylindric Algebra ♦ First-order Variable-assignment Tuples ♦ Usual Boolean Operator ♦ Kripke Frame ♦ Several Possible Connection ♦ Modal Perspective ♦ Propositional Multi-modal Logic ♦ Two-dimensional Case ♦ Propositional Multi-modal Language Ml ♦ First-order Quantification Vi ♦ First-order Formula ♦ Basic Notion ♦ Classical First-order Logic ♦ Multi-dimensional Modal Formalism ♦ Axiomatisation Question ♦ Modal Logic ♦ Kripke Semantics ♦ Unary Modality ♦ Coordinate-wise Modal Operator ♦ Two-dimensional Propositional Modal Logic ♦ Possible World Abstract Abstract. We analyse the role of the modal axiom corresponding to the first-order formula “∃y (x = y) ” in axiomatisations of two-dimensional propositional modal logics. One of the several possible connections between propositional multi-modal logics and classical first-order logic is to consider finite variable fragments of the latter as ‘multi-dimensional ’ modal formalisms: First-order variable-assignment tuples are regarded as possible worlds in Kripke frames, and each first-order quantification ∃vi and ∀vi as ‘coordinate-wise ’ modal operators ✸i and ✷i in these frames. This view is implicit in the algebraisation of finite variable fragments using finite dimensional cylindric algebras [6], and is made explicit in [15, 12]. Here we look at axiomatisation questions for the two-dimensional case from this modal perspective. (For basic notions in modal logic and its Kripke semantics, consult e.g. [2, 3].) We consider the propositional multi-modal language ML δ 2 having the usual Boolean operators, unary modalities ✸0 and ✸1 (and Educational Role Student ♦ Teacher Age Range above 22 year Educational Use Research Education Level UG and PG ♦ Career/Technical Study Learning Resource Type Article